Inverse Laplace transform
 Inverse Laplace transform

Mellin's inverse formula
An integral formula for the inverse Laplace transform, called the Bromwich integral, the Fourier–Mellin integral, and Mellin's inverse formula, is given by the line integral:
where the integration is done along the vertical line Re(s) = γ in the complex plane such that γ is greater than the real part of all singularities of F(s). This ensures that the contour path is in the region of convergence. If all singularities are in the left halfplane, or F(s) is a smooth function on  ∞ < Re(s) < ∞ (i.e. no singularities), then γ can be set to zero and the above inverse integral formula above becomes identical to the inverse Fourier transform.
In practice, computing the complex integral can be done by using the Cauchy residue theorem.
It is named after Hjalmar Mellin, Joseph Fourier and Thomas John I'Anson Bromwich.
Post's inversion formula
An alternative formula for the inverse Laplace transform is given by Post's inversion formula.
See also
 Inverse Fourier transform
References
 Davies, B. J. (2002), Integral transforms and their applications (3rd ed.), Berlin, New York: SpringerVerlag, ISBN 9780387953144
 Manzhirov, A. V.; Polyanin, Andrei D. (1998), Handbook of integral equations, London: CRC Press, ISBN 9780849328763
External links
This article incorporates material from Mellin's inverse formula on PlanetMath, which is licensed under the Creative Commons Attribution/ShareAlike License.
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